A259472 Coefficients in an asymptotic expansion of A003319(n)/n! in falling factorials.

1, -2, -1, -4, -19, -110, -745, -5752, -49775, -476994, -5016069, -57462828, -712732987, -9521244982, -136356161873, -2084860795232, -33907076207495, -584602069590058, -10652917092110429, -204604743619641620, -4131502481607654739, -87507494737954740126

Offset

0,2

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..446

L. Comtet, Sur les coefficients de l'inverse de la série formelle Sum n! t^n, Comptes Rend. Acad. Sci. Paris, A 275 (1972), 569-572.

L. Comtet, Series inversions, C. R. Acad. Sc. Paris, t. 275 (25 septembre 1972), 569-572. (Annotated scanned copy)

R. K. Guy, Letter to N. J. A. Sloane, Mar 1974

Formula

From Vaclav Kotesovec, Aug 12 2015: (Start)

G.f.: (1/Sum(k! x^k))^2.

Expansion of (1-g(x))^2, where g(x) is the g.f. of A003319.

a(n) ~ -2*n! * (1 - 3/n - 4/n^3 - 33/n^4 - 283/n^5 - 2785/n^6 - 31291/n^7 - 395360/n^8 - 5544754/n^9 - 85427259/n^10), for coefficients see A261214.

For n>0, a(n) = A059439(n) - 2*A003319(n).

For n>0, a(n) = Sum_{k=1..n} A260503(k) * Stirling1(n-1, k-1).

(End)

Example

A003319(n) / n! ~ 1 - 2/n - 1/(n*(n-1)) - 4/(n*(n-1)*(n-2)) - 19/(n*(n-1)*(n-2)*(n-3)) - 110/(n*(n-1)*(n-2)*(n-3)*(n-4)) - 745/(n*(n-1)*(n-2)*(n-3)*(n-4)*(n-5)) -  ... [coefficients are A259472]
A003319(n) / n! ~ 1 - 2/n - 1/n^2 - 5/n^3 - 32/n^4 - 253/n^5 - 2381/n^6 - ... [coefficients are A260503]

Mathematica

CoefficientList[Series[1/Sum[k! * x^k, {k, 0, 20}]^2, {x, 0, 20}], x] (* Vaclav Kotesovec, Aug 03 2015 *)
CoefficientList[Assuming[Element[x, Reals], Series[E^(2/x) * x^2 / ExpIntegralEi[1/x]^2, {x, 0, 25}]], x] (* Vaclav Kotesovec, Aug 03 2015 *)

Crossrefs

Cf. A003319, A260503, A261214, A261239, A261253, A261254, A059439.

Sequence in context: A013162 A010252 A032105 * A354055 A375354 A053565

Adjacent sequences: A259469 A259470 A259471 * A259473 A259474 A259475

Keyword

sign

Author

N. J. A. Sloane, Jul 03 2015, following a suggestion from R. K. Guy, Apr 29 1974

Extensions

More terms from Vaclav Kotesovec, Aug 01 2015

New name from Vaclav Kotesovec, Aug 12 2015

Entry revised by Vaclav Kotesovec, Aug 12 2015

Status

approved